A linear space is an incidence structure of points and lines such that every pair of points is contained in a unique line. In the first two chapters of this thesis results are presented linking structural properties to arithmetic conditions on the number of points and lines. We provide a short new proof of Jim Totten's classification of all linear spaces for which the difference between the number of points and lines does not exceed the square root of the number of points. We extend this classification when the number of points is of a certain form. Also in these chapters we have similar classification results for more specialized finite geometrical structures such as (r,l)-designs.
The last chapter is devoted to (k,u)-arcs. A (k,u)-arc in a finite projective plane is a set of k points meeting no line of the plane in more than u points. Elementary bounds upon k can be established and we call an arc with this maximum number of points perfect. An arc not properly contained in any other is called complete. Several constructions are given for both perfect and complete arcs. The major results of this chapter concern the uniqueness of completions of a (k,u)-arc to a perfect arc.
